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An analytical study is carried out on the cut-off frequencies of Lamb waves in freestanding thin films made of various functionally graded elastic, piezoelectric, or piezoelectric–piezomagnetic materials. Results show that the set of cut-off frequencies is a union of two series of approximate arithmetic progression, in which the differences are inversely proportional to the definite integral of a function of the material parameters along thickness. Given the simple and universal relationship between cut-off frequencies and material parameters, this study provides theoretical guidance not only for nondestructive evaluation in engineering applications but for designing high-performance sensors based on Lamb waves.